proceedings of the
american mathematical society
Volume 114, Number 2, February 1992
THE FEKETE-SZEGO PROBLEM
FOR STRONGLY CLOSE-TO-CONVEX FUNCTIONS
H. R. ABDEL-GAWADAND D. K. THOMAS
(Communicated by Clifford J. Earle, Jr.)
Abstract.
Let K(ß) denote the class of normalized analytic strongly close-toconvex functions of order ß > 0 , defined in the unit disc D and let / e K(ß),
with f(z) = z + a2z2 + a^z3 H-,
for z £ D . Sharp bounds are obtained for
|fl3 - pa\\ when p. is real.
Introduction
Denote by S the class of normalized analytic univalent functions / defined
for z G D = {z : \z\ < 1} by
(1)
f(z) = z + Ya"z"«=2
A classical theorem of Fekete and Szegö [2] states that for f e S given by (1),
{3 - 4p,
if p < 0
l + 2e-2^1-"),
ifO<p<l
4p-3,
ifp>l.
This inequality is sharp in the sense that for each p there exists a function in
S such that equality holds. Recently Pfluger [8] has considered the problem
when p is complex. In the case of C, S* and K, the subclasses of convex,
starlike and close-to-convex functions respectively, the above inequalities can be
improved [5, 6]. In particular for / e K and given by (1), Keogh and Merkes
[5] showed that
|c23- pa\\
< <
3-4p,
l/3 + 4/9p,
1,
4p-3,
ifp<l/3,
ifl/3<p<2/3,
if 2/3 < p < 1,
ifp>l.
Again, for each p, there is a function in K such that equality holds. In this
paper we extend this result to the class K(ß) of strongly close-to-convex functions of order ß in the sense of Pommerenke [9]. Thus / G K(ß) if, and
Received by the editors March 9, 1989.
1980 Mathematics Subject Classification (1985 Revision). Primary 30C45.
©1992 American Mathematical Society
0002-9939/92 $1.00+ $.25 per page
345
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H. R. ABDEL-GAWADAND D. K. THOMAS
346
only if, /, given by (1), is analytic in D and is such that there exists g e S*
satisfying
arg
(2)
zf'(z)
nß
<
g(z)
for z G D and ß > 0. Clearly K(0) = C, K(l) = K and when 0 < ß < 1 ,
K(ß) is a subset of K and hence contains only univalent functions. However
in [4], Goodman showed that K(ß) can contain functions with unbounded
valence for ß > 1.
Recently, Koepf [7] has considered the Fekete-Szegö problem for K(ß) and
obtained sharp results for some particular values of p, all of which, with the
exception of the case p = 1 and ß > 1 , are contained in the following result.
Results
Theorem. Let f e K(ß) and be given by (I). Then for 0 < ß < 1,
1 -p + ß(2-3p)(ß
\cii-pa¡\
2ß
+ 2)
21
/î(2-3p)2
^+
3
+
3[2-/?(2-3p)]
< {
2^+1
p- 1 +
ifß< 3(jff+ l)'
2ß
if 3(/? + l) <ß<
-fi-
2
3'
2(ß + 2
^"¿M
+ iy
2Q8+ 2)
ifß > 3(^ + 1)
/?(3p - 2)(/? + 2)
whilst for ß > I, the first two inequalities hold. For each p there are functions
in K(ß) such that equality holds in all cases.
We shall require the following:
Lemma 1 ([10, p. 166]). Let h e P, i.e., let h be analytic in D and satisfy
Reh(z) > 0 for z e D, with h(z) = 1 +cxz + c2z2 + ■■■, then
Cl
<2-^.
Lemma 2 ([6, Lemma 3]). Let g e S* with g(z) = z + 62z2 + ¿3z3 H-,
then
for p real,
\b3-pb22\<max{l,\3-4p\}.
We note that Lemma 2 above can easily be extended to the wider class S* (a)
of strongly starlike functions of order a > 0, i.e., g analytic and normalized
in D and satisfying
arg
zg'(z)
an
g(z)
see e.g. [1]. In this case, one obtains the sharp inequality
|Z?3- p¿b2| < maxja,
a2|3 - 4p|},
for p real.
Proof of Theorem. It follows from (2) that we can write
(3)
zf'(z) = g(z)h(z)B
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THE FEKETE-SZEGÖPROBLEM FOR STRONGLY CLOSE-TO-CONVEXFUNCTIONS
347
for g e S* and h e P. Equating coefficients in (3) we obtain 2a2 = ßcx + b2
-1 + ßc2 + ßcxb2 + b3, so that
and 3a3 = (ß(ß - l)/2)c2.2
phi 1+ 4 I c2+ 0(2-3p)
1
fl3 - p£2|
1\ ,
2'Cl
(4)
We consider first the case (2ß)/(3(ß
|tf3-pa||
< -
b3-fP.bj
<l-p+y(
+ 1)) < p < 2/3. Equation (4) gives
+
C2"2Cf
02(2-3p), ,
12
c?|+^Q-f)|c,||H
1
02(2-3p),„2l
12
5KÎ +
ß
+, yg(2-3p)
■hi,
= 0(x) say, with x = \cx\,
where we have used Lemmas 1 and 2 and the fact that \b2\ < 2 for g e S*.
An elementary argument shows that the function <P attains a maximum at
x0 = 2(2 - 3/0/(2 - 0(2 - 3p)), and so
\a3-p.aj\
<<P(x0),
which proves the theorem if p < 2/3 and ß >0.
2(2-3ji)
c\
c2 = 2,
2-/?(2-3p)'
in (4) shows that the result is sharp.
Choosing
b2 = 2,
and
63 = 3,
We note that \cx\ < 2, that is, p >
2ß/(3(ß + l)).
Next consider the case p < (2ß)/(3(ß
\a3-pai\
< a3
2ß
3(0+1
A+
2ß
,l +f + 3(0+1)
+ 1)). Then
20
3(0+1)
ß)(ß
- p
+ l)2
\a2
1 -p + 0(2-3p)(0 + 2)
for 0 > 0, where we have used the result already proved in the case p =
20/3(0 + 1), and the fact that for / g K(ß), the inequality \a2\ < 0 + 1 holds
[3]. Equality is attained on choosing X = 0, px = p2 = b2 = 2, and ¿3 = 3 .
Suppose now that 2/3 < p < (2(0 + 2))/(3(0 + 1)). Since gES'
we can
write zg'(z) = g(z)p(z) for p G P, with p(z) — I + pxz +P2Z2 + ••• , and so
equating coefficients we have that 62 = px and 2Z>3= p2 + p2.
We deal first with the case p = 2(0 + 2)/(3(0 + 1)). Thus (4) gives
2(0 + 2) 2 _ 1 /
fl3-370TTf2-6"r-T
p
02c?
+f I
ßP\C\
6(0 + 1) 3(0+ i;
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+
0-1
12(0+1 ;Pi
348
H. R. ABDEL-GAWADAND D. K. THOMAS
and so if 0 < 1,
a3
2(0 + 2)
3(0 + 1)'
< 1
Pi +
6 P2
2\/2\
ß2\c
+
<
1
1-0
+ 12(0+1) \P
\PÎ\ + 0
3
2
2|„2|
ßl\c
+ 6(0 + 1
20+1
3
< 20+1
(1
12(0 +^Wl
1)
+ fi\PlCl\
3(0 + 1)
6(0 +
6
+
Cl
+
0|PlCl|
3(0+ 1]
ß
6(0 + 1 (\ct\-\Pi
where we have used Lemma 1.
Now write
«3
ml
(0+l)(3p
2(0 + 2)
3(0+1)
3(0 + 1) /2(0 +2)
W
2)
«3
+—r- I370TTJ
-n r
tf
and the result follows at once on using the theorem already proved in the cases
p = 2/3 and p = 2(0 + 2)/(3(0 + 1)) for 0 < 1 . Equality is attained when /
is given by
f'tz) = (1
(l +
^z z2)"
>
J { '
(1-z2)^1'
We finally assume that p > (2(0 + 2))/(3(0 + 1)). Write
'2(0 + 2)
2(0 + 2) 2
«3 - ßa2
a3
3(0+1)■a2V
V3(0+l)
a2,
and the result follows at once on using the theorem already proved for p =
2(0 + 2)/3(0 +1)) in the case 0 < 1 and the inequality \a2\ < 0 + 1 , which
was proved in [3]. Equality is attained in this last case on choosing cx - b2 = 2/,
c2 = -2 and è3 = -3 in (4).
We remark that the methods used in [5, 6], together with equation (4), suggest
that in order to obtain sharp results for 0 > 1 and p > 2/3 , an extension to the
"area principle" may be required. Since K(ß) contains functions of unbounded
valence for 0 > 1 establishing sharp estimates in this case may require deeper
methods.
References
1. D. A. Brannan and W. E. Kirwan, On some classes of bounded univalent functions, ¡. London
Math. Soc. (2) 1 (1969), 431-443.
2. M. Fekete and G. Szegö, Eine Bermerkung über ungerade schlichte Funktionen, J. London
Math. Soc. 8(1933), 85-89.
3. A. W. Goodman,
On close-to-convex functions of higher order, Ann. Univ. Sei. Budapest.
Eötvös Sect. Math. 15 (1972), 17-30.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
THE FEKETE-SZEGÖPROBLEM FOR STRONGLY CLOSE-TO-CONVEXFUNCTIONS
4. _,
349
A note on the Noshiro-Warschawski theorem, J. Analyse Math. 25 (1972), 401-408.
5. F. R. Keogh and E. P. Merkes, A coefficient inequality for certain classes of analytic functions,
Proc. Amer. Math. Soc. 20 (1969), 8-12.
6. W. Koepf, On the Fekete-Szegö problem for close-to-convex functions, Proc. Amer. Math.
Soc. 101 (1987), 89-95.
7. _,
On the Fekete-Szegö problem for close-to-convex functions. II, Arch. Math. 49 (1987),
420-433.
8. A. Pfluger, The Fekete-Szegö inequality for complex parameters, Complex Variables Theory
Appl. 7(1986), 149-160.
9. Ch. Pommerenke,
On close-to-convex analytic functions, Trans. Amer. Math. Soc. 114
(1965), 176-186.
10. _,
Department
Univalent functions, Vandenhoeck and Ruprecht, Göttingen,
of Mathematics
and Computer
Science,
University
Swansea SA2 8PP, Wales
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1975.
College
Swansea,